Price Evolution in a Continuous Double Auction Prediction Market

Price Evolution in a Continuous Double Auction Prediction Market With a
Scoring-Rule Based Market Maker⇤
Mithun Chakraborty, Sanmay Das, Justin Peabody
{mithunchakraborty,sanmay}@seas.wustl.edu, [email protected]
Department of Computer Science and Engineering, Washington University in St. Louis
Abstract
The logarithmic market scoring rule
(LMSR), the most common automated
market making rule for prediction markets,
is typically studied in the framework of
dealer markets, where the market maker
takes one side of every transaction. The
continuous double auction (CDA) is a much
more widely used microstructure for general
financial markets in practice. In this paper,
we study the properties of CDA prediction
markets with zero-intelligence traders in
which an LMSR-style market maker participates actively. We extend an existing idea of
Robin Hanson for integrating LMSR with
limit order books in order to provide a new,
self-contained market making algorithm that
does not need “special” access to the order
book and can participate as another trader.
We find that, as expected, the presence of
the market maker leads to generally lower
bid-ask spreads and higher trader surplus (or
price improvement), but, surprisingly, does
not necessarily improve price discovery and
market efficiency; this latter effect is more
pronounced when there is higher variability
in trader beliefs.
Introduction
Financial markets, such as those for stocks, bonds, and
options, provide participants with opportunities for hedging, investment, and speculation. Prediction markets (which
can be thought of as a type of binary option) are often
used to forecast future events like elections or sports outcomes, with participants using either real or play money
[Berg and Rietz, 2006; Cowgill and Zitzewitz, 2014]. In
order to deal with the chicken-and-egg problem of liquidity, many markets employ market makers, specially designated agents that are responsible for providing liquidity
by always being ready to transact with traders. In the last
decade, research on algorithmic market making has become
⇤
This research was partially supported by an NSF CAREER
award (1414452).
Copyright c 2015, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.
one of the interesting contact points between artificial intelligence and finance, both in the general context [Das, 2005;
2008; Wah and Wellman, 2014], and specifically in the design of prediction markets ([Hanson, 2003b; Chen, 2007;
Brahma et al., 2012; Othman et al., 2013; Abernethy et al.,
2014] etc.).
Most research on algorithmic market making in both financial and prediction markets has either focused on market
making as a trading strategy [Chakraborty and Kearns, 2011;
Schmitz, 2010] or has modeled the market as a pure dealer
market, where the market maker takes one side of every
trade [Hanson, 2003b; Das, 2008; Othman et al., 2013]. The
market can therefore be modeled in terms of the market
maker’s quoted bid (buy) and ask (sell) prices, and traders’
decisions on whether or not to transact at these prices. However, most modern markets, ranging from big financial markets like the NYSE and NASDAQ to smaller prediction markets like the Iowa Electronic Markets, use the continuous
double auction (CDA) mechanism [Forsythe et al., 1992].
In CDAs, participants can place limit orders that specify a
transaction price and are guaranteed to only execute at that
price or better (although execution is, of course, no longer
guaranteed). The key element of CDAs is the limit order
book, which contains all active buy and sell limit orders; the
highest buy and the lowest sell constitute the market bid and
ask prices at any point in time.
While most practical market making algorithms (for example, those used by market makers on the NYSE and NASDAQ) are deployed in markets with limit order books, the
academic literature on algorithmic market making has thus
far produced almost no analysis of the impact of market
making in CDA markets (with the exception of [Wah and
Wellman, 2014]). Here we begin to tackle this problem in
the context of market making in prediction markets. The
logarithmic market scoring rule proposed by Robin Hanson [2003b] is probably the most commonly deployed automated market maker in prediction markets. Hanson [2003a]
also provides a scheme for integrating order books with his
market making algorithm which, to the best of our knowledge, has not yet been evaluated in the literature. This
scheme, as proposed, involves the market maker having special access to orders before they hit the order book, and a
“parallel” implementation that looks at the incoming order,
the order book, and executes portions of the trade with the
market maker and portions with the existing orders on the
order book. In addition to the special system privileges this
requires, it is not entirely transparent to traders, since the order books themselves never reflect the market maker’s presence (and thus give a worse impression of the state of prices
and the bid-ask spread than reality).
In this paper, we propose a modification of Hanson’s
scheme for integrating LMSR with CDA mechanisms that
allows an LMSR-based market making agent to compute
limit bid and ask prices and participate in the order books as
any other trader would, while still maintaining the key desirable properties – namely improved liquidity with bounded
worst-case loss. We call this the “Integrated” market maker
(as opposed to the “Parallel” market maker of the original
scheme). In general, analysis of the properties of market
making algorithms in practice is difficult, since they affect
the dynamics of the pricing mechanism itself, and therefore the standard practice of backtesting on historical data
is of very limited value. However, there is evidence that
simulation models with zero-intelligence (ZI) traders [Gode
and Sunder, 1993] can replicate many key features of limit
order book dynamics [Farmer, Patelli, and Zovko, 2005;
Othman, 2008] and have practical value in assessing the
properties of market making algorithms [Brahma et al.,
2012]. Therefore, we evaluate market properties in prediction markets populated by ZI traders; we compare the parallel and integrated implementations of LMSR with a situation
where no market maker is present, and also a pure dealer
market mediated by LMSR. We are mainly interested in the
following properties:
• Information aggregation properties: For example, how
fast does the market price converge to the true underlying asset value? How far away is the price from the true
value, on average?
• Market quality properties: For example, how liquid is the
market, as measured by the bid-ask spread? How much
surplus or price improvement does the market generate?
In our experiments we find that the presence of the market
maker leads to generally lower bid-ask spreads and higher
trader surplus (or price improvement), but, surprisingly, does
not necessarily improve price discovery and market efficiency; this latter effect is more pronounced when there is
higher variability in trader beliefs.
Market Model
In this section we describe the precise market model we
use and the algorithms used for trading and market making. We simulate four different market microstructures: (1)
A Continuous Double Auction (CDA) mechanism without
any market maker (pureCDA); (2) A CDA with the “Integrated” implementation of LMSR (INT); (3) (2) A CDA
with the “Parallel” implementation of LMSR (PAR); (4) A
pure dealer framework with all trades going through a traditional LMSR market maker (pureLMSR).
Prediction market We focus on a prediction market set
up to forecast whether a single extraneous uncertain event,
which can be modeled as a binary random variable X, will
occur at some pre-determined future date; on that date, the
market terminates, and every unit (share) of the asset traded
in the market is worth $1 if the event occurs and is worthless otherwise; we call this cash equivalent of the asset its
liquidation value or true value. Before that date, anyone can
place orders to buy or (short-)sell any amount of the asset in
the market at prices in the interval [0, 1], i.e. the market institution does not impose any budget constraints on traders.
We also assume that there is a fixed probability distribution
with Pr(X = 1) = ptrue from which the realization of X
is drawn on the termination date so that the expected “true”
value of the asset is ptrue , but no agent in the world knows
this ptrue precisely.
Types of orders Traders in a financial exchange can typically place buy/sell orders of two kinds: (1) market orders
that specify only a quantity and demand immediate execution, hence accept any price offered by the other party, and
(2) limit orders that specify both a quantity and a limit on acceptable transaction prices (called a limit price or marginal
price) but are not guaranteed execution. A Continuous Double Auction (CDA) maintains two order books, one for buy
orders (bids) and the other for sell orders (asks), which are
two priority queues for outstanding limit orders prioritized
by limit price and arrival time (higher priority is assigned to
a buy order with a higher bid price and a sell order with a
lower ask price). Any incoming limit order is placed on the
appropriate book, and the mechanism automatically checks
to see if the current best (highest) bid is at least as large as
the current best (lowest) ask; if yes, then the smaller of the
two quantities ordered is traded at the limit price of the order
that arrived earlier, the books are updated, and this is continued till the best ask exceeds the best bid. Any new market
order is executed immediately, perhaps partially, against the
best available outstanding order(s) or is rejected if the book
on the other side is empty. In our simulations, all traders
place limit orders only but some of them can become market orders effectively, e.g. if an incoming limit buy order
“crosses” the books. i.e. its bid is no less than the best ask(s)
on the sell order book, and its demand does not exceed the
supply of said booked order(s).
Logarithmic Market Scoring Rule We now briefly describe the LMSR market maker for a single-security prediction market liquidating in {0, 1} [Hanson, 2003b; Chen,
2007]. Its “state” is described by a real scalar qmm , interpreted as the net outstanding quantity of the security; its
instantaneous price at this state, i.e. cost per share of buying/selling an infinitesimal amount from/to LMSR, is given
eqmm /B
by pmm = 1+e
qmm /B where B > 0 is a parameter controlling all properties of the market maker. A trader placing a market order for buying any finite quantity Q of assets from LMSR would
have to ⌘
pay it a dollar amount
⇣
1+e(qmm +Q)/B
C(qmm ; Q) = B ln
and after the transac1+eqmm /B
tion, the market maker’s state is updated to (qmm + Q); for a
sell order, the same formula applies by setting Q to the nega-
tive of the supplied quantity, and C(qmm ; Q) > 0 becomes
the sales proceeds. One key property of LMSR is that it’s
loss is bounded (for the binary case by B ln 2).
Population of traders Every agent other than the market
maker is called a “background” trader [Wah and Wellman,
2014]. Before every simulation, the expected true asset value
ptrue is chosen at random from a common-knowledge common prior which is a uniform distribution on [0, 1]. Every
trader i then observes a private sequence of Ntrials Bernoulli
trials with probability of success ptrue , and sets her idiosyncratic valuation of the asset to her Bayesian posterior expeci +1
tation of the true value, vi = Nxtrials
+2 where xi is the number
of successes in her sample. Thus, Ntrials is a measure of the
precision of the signal that each trader receives, related to
the inverse of the variance of beliefs across the population,
similar to the model of Zhang et al [2012]. The implementation of a trading decision on top of the belief then follows the
zero-intelligence (ZI) trader model [Gode and Sunder, 1993;
Othman, 2008], with the addition of non-unit trade sizes. At
each step of a simulation (a “trading episode”), a trader is
picked uniformly at random and is assigned buyer or seller
status with equal probability except for pureLMSR (see below). She then places her limit order, the limit price being
drawn uniformly at random from [vi , 1] if she is a seller and
from [0, vi ] if she is a buyer, and the order quantity from a
common exponential distribution with mean = 20 which
is known to the market mechanism.
(1) pureCDA We have already fully explained the interaction between a CDA mechanism with no market making and
the trading population under Types of Orders.
(2) PAR The parallel implementation is a single-security
version of Robin Hanson’s “booked orders for market scoring rules” [Hanson, 2003a]. We delineate its operation for
a buy order, the treatment of sell orders being symmetric.
Suppose a limit buy order for a quantity qb at a limit price
(bid) pb arrives when the LMSR market maker’s instantaneous price is pmm , and the current best bid and ask prices
are bmax , amin (at market inception, both books are empty,
and pmm = 0.5). If pb  pmm 1 , the order cannot be immediately executed, so it is pushed on to the limit buy order book. If pmm < pb , and qb is not large enough to drive
pmm beyond min{pb , amin }, then the incoming order is completely executed with the market maker according to the traditional LMSR algorithm; otherwise, if pmm < pb < amin ,
it is only partially executed with LMSR till pmm reaches pb ,
the residual order being placed on the buy order book; but if
pb
amin , LMSR sells only till its instantaneous price hits
amin after which the incoming order executes against the best
booked ask. If the top level of the book is exhausted but the
incoming order is not, LMSR is invoked again, and this process recurs till either the order is finished or the new best ask
exceeds the order’s bid price. The loss bound of the standard
LMSR algorithm is maintained in this case.
1
In this implementation, pmm always lies between the best ask
and bid prices on the books, so pb  pmm implies that pb does not
exceed the minimum ask price either.
(qa2, a2)
(qa1, a1)
(qa2, a2)
(qa1, a1)
(qask, amm)
Purchase of qask at average price amm
from LMSR required for this jump
Sale of qbid at average price bmm to LMSR
required for this jump
pmm
(qbid, bmm)
(qb1, b1)
(qb2, b2)
(qb1, b1)
(qb2, b2)
Case 3.
(qa2, a2)
(qa1, a1)
(qa2, a2)
(qa1, a1)
(qask, amm)
(qa2, a2)
(qa1, a1)
(qb1, b1)
(qb2, b2)
(qb1, b1)
(qb2, b2)
(qb1, b1)
(qb2, b2)
pmm
(qa2, a2)
(qa1, a1)
(qbid, bmm)
(qb1, b1)
(qb2, b2)
pmm
Case 1.
Case 2.
Figure 1: Illustration of how the market maker in the INT
setting places ask and bid quotes every time the state of the
books changes.
(3) INT In this novel “integrated” implementation that we
propose, whenever the best ask and bid prices on the books
change, an LMSR-based agent steps in.
1. If its instantaneous price pmm  bmax , then LMSR generates⇣ only a ⌘limit sell order for a quantity
qask
⇣
⌘ =
1/pmm 1
1 pmm
B
B ln 1/amin 1 at an ask price of qask ln 1 amin .
2. If pmm
amin ,⌘then it generates a⇣buy order
⇣
⌘ for qbid =
1/bmax 1
1 bmax
B
B ln 1/pmm 1 at a bid of qbid ln 1 pmm .
3. If bmax < pmm < amin , both orders are generated.
Note that if fully executed immediately these orders would
take the LMSR price to bmax and amin respectively. The
LMSR trader then replaces all its earlier orders with the new
order(s) if this action does not immediately cross the books,
otherwise it sits idle. After this step, the market is now
ready to accept a new order from the background traders,
or continue the execution of a partially filled outstanding
order, as the case may be. Thus, this market maker can be
implemented in practice as just another trader, which is a
significant benefit over the PAR framework where the market maker requires some special access to incoming trades
and order books. Moreover, any feasible trade with the INT
market maker is executed at its actual quoted price rather
than following the non-linear LMSR pricing function, which
makes trading more transparent and intuitive to traders.
The original LMSR loss bound again holds. Also, we can
prove that INT myopically imposes at least as high a cost
on the next arriving trader as PAR, assuming that the market
makers and order books are in the same state.
Proposition 1. Suppose the LMSR market maker in both
PAR and INT are in state q, and the order books are also
otherwise identical. For any next arriving trade, the immediate cost incurred by the next trader is at least as high for
INT as it is for PAR.
Proof Sketch. Consider the last of the three cases for INT
above, bmax < pmm < amin , and let Q⇤ be the quantity one
would need to buy from LMSR to bring its price to amin .
Then, if the current state of the INT market maker is q, it
⇤
)
will place a sell order of Q⇤ at an ask of C(q;Q
. Now if a
Q⇤
buy order for Q < Q⇤ arrives with a sufficiently high bid,
the whole of it will execute with the market maker,
and the
⇤
)
immediate earnings of the latter will be QC(q;Q
.
If
the
PAR
Q⇤
market maker had the same state q (hence the same pmm )
when the same buy order arrived, the ensuing trade would
cost the trader C(q; Q) which is less than INT’s earnings
⇤
)
since C(q;Q)
< C(q;Q
from the convexity of C. Similar
Q
Q⇤
arguments apply to the other cases.
This result suggests that INT might provide somewhat
less liquidity in general than PAR, and incur less loss in doing so, but we do not expect them to be very different. However, this is a loose prediction, since the result is myopic –
it says nothing about price evolution in a market; given the
market maker’s active role, the dynamics of the evolution of
q and the order book could conceivably end up quite different. We examine this issue further in the experiments.
(4) pureLMSR In this setting, traders still place limit orders
but an LMSR market maker takes one side of every trade.
At each trading episode, a trader arrives and compares her
private valuation vi to the current market price pmm . If vi >
pmm , she decides to buy; if vi < pmm , she decides to sell, and
leaves without placing any order otherwise. Then she picks
her limit price and order size exactly as the ZI traders above.
The quantity bought/sold is the minimum of the order size
and the quantity needed to drive the LMSR’s instantaneous
price to the trader’s limit price, and monetary transfers are
determined by the above function C(·; ·).
Note that all components of each limit order of a trader are
independent of the market state for all four settings, except
for the direction of the trade (buy/sell) in pureLMSR.
Evaluation
We present an overview of the various measures we use to
evaluate the properties of our market environments.
Information aggregation properties:
• ConvTime (Convergence time): This is defined as the
number of trading episodes it takes for the “market price”
pM to get within a band of size ±0.05 around the true
expected asset value ptrue for the first time; pM (t) is measured at the end of every trading episode t as the mid-point
of the bid-ask spread ((bmax (t) + amin (t))/2) for each of
the models with CDA, and as the LMSR instantaneous
price for the pure dealer case.2 Thus,
ConvTime = min{t : pM (t) 2 [ptrue 0.05, ptrue +0.05]}.
A lower convergence time means that the market’s esti2
If the market price does not enter this band over the duration
of the simulation, ConvT ime is set to ntrades = 500; in our simulations, this is rarely observed.
mate (price) quickly gets close to the true expected asset
value, i.e. the market is efficient.
• RMSD and RMSDeq : RMSD is the root-mean-squared
deviation of the market price (defined above) from
ptrue over the entire simulation (ntrades trading episodes).
RMSDeq is the root-mean-squared deviation between the
same quantities but over only the “equilibrium period”,
i.e. for t
ConvTime. Lower values of these measures
indicate lower price volatility, another desirable property
from an information aggregation perspective.
Market quality properties:
• Spread and Spreadeq : For each scenario with a CDA, the
market bid and ask prices bM (t) and aM (t) at the end of
each trading episode are the highest bid bmax and the lowest ask amin on the books respectively (set to 0 and 1 if
the corresponding book is empty). For the pure dealer setting, we assume that the market maker knows the average
order size of the trading population, so for a current
market state of qmm , the effective market quotes are taken
to be aM = C(qmm ; ) and bM = C(qmm ; ) which are the
prices per share of buying and selling shares from and
to LMSR at the current state respectively. In our notation,
“Spread” denotes the bid-ask spread (aM (t) bM (t)) averaged over all ntrades episodes, while “Spreadeq ” is the
average taken over the equilibrium period only, as above.
The bid-ask spread is widely used as a proxy for market
liquidity and smaller values are better, since they imply
lower trading costs.
• (Idiosyncratic) TraderSurplus: If a trader with idiosyncratic valuation v places a buy order of which a quantity q
goes through at an execution price pexec , then the trader’s
surplus is defined as q(v pexec ) (similarly, a seller’s surplus is q(pexec v)). TraderSurplus denotes the sum of
individual surpluses of all background traders. Also note
that (v pexec ) and (pexec v) correspond loosely to the
notion of price improvement, when weighted by the probability of execution at that difference. So, even in settings
where the private or idiosyncratic value assumption is untenable, the surplus is still a useful measurement of how
much value participants are getting from being in one particular microstructure over another. Since every order executes at a price at least as desirable as its limit price, all
trader price improvements (surpluses) are positive.
• MMloss: This is the loss incurred by the market making
mechanism, computed just like (the negative of) a trader
surplus, with the private valuation replaced with the true
expected asset value ptrue . Obviously, this does not apply to pureCDA. Since the market is an ex post zero-sum
game between the market maker and the trading population, this measure is also numerically equal to the true
expectation of the traders’ collective net payoff. This measure is particularly important when the market institution
itself subsidizes the market maker.
Results
We ran three sets of 1000 simulations each. In each set, we
used a different value of the parameter Ntrials (20, 40, 100)
(a) ConvTime
(c) RMSDeq
(b) RMSD
250
pureCDA
INT
PAR
200
9
8
8
7
7
pureLMSR
6
6
5
150
5
4
4
100
3
3
2
2
50
1
1
0
20
40
100
0
20
100
0
20
(e) Spreadeq
(d) Spread
5
pureCDA
INT
PAR
pureLMSR
40
40
100
(f) Vol
3.5
1000
4.5
900
3
4
800
2.5
3.5
3
700
600
2
2.5
500
1.5
2
1.5
400
300
1
1
200
0.5
0.5
0
100
20
40
100
0
20
(g) Vol*
40
100
0
20
(h) TraderSurplus
40
100
(i) MMloss
1000
90
20
900
80
18
800
70
16
pureCDA
INT
PAR
pureLMSR
700
14
60
600
12
50
500
10
40
400
8
30
300
6
200
20
4
100
10
2
0
0
20
40
100
20
40
100
0
20
40
100
Figure 2: Experimental results, averaged over 1000 simulations each. The labels along the horizontal axis indicate the number
of private Bermoulli trials with success probability ptrue observed by each trader in the respective simulation set; this number is
directly related to the precision in trader beliefs. Values along vertical axis units are in cents in panels (c)-(f) and in dollars in
(h), (i).
controlling the precision of trader beliefs. In each simulation, we made the same random sequence of ntrades = 500
traders interact with each of our four microstructures. The
LMSR parameter B is fixed at 100 for all simulations. We
computed all of the above measures for each simulation, and
then averaged them over all 1000 simulations. The results
are presented in Figure 2, and the analysis follows. Note that,
the values (rmsd of prices, spreads) depicted in Figures 1(c)(f) are in cents while those in the last two figures (surplus,
losses) are in dollars, for clarity.
Information aggregation: ConvTime (a) follows the pattern: pureLMSR << pureCDA < INT < PAR. However, in
terms of stability (RMSD, overall (b) and in equilibrium (c)),
pureCDA fares the best and the two hybrid mechanisms are
very close to each other. The quick convergence and high
volatility of LMSR are well-known; surprisingly, coupling
it with a CDA delays convergence drastically, but it does
ensure more stable prices (lower RMSDeq ) once the price
converges. While it seems that the market maker-CDA combination might impede the market’s learning abilities, it is
likely in this case to be an artifact of the fixed beliefs held
by ZI traders, who stick to their beliefs no matter what happens to the price – it’s not clear that any scoring rule style
of market maker would be able to learn quickly when the
signals have high variance and the traders don’t update their
signals. This hypothesis is borne out by the fact that the effect diminishes as the variance in traders’ beliefs decreases.
Liquidity / Trading activity: Perhaps the biggest reason to
deploy a market-maker is to reduce spreads. Figures 1 (d)
and (e) show that INT serves this purpose more effectively
than pureCDA. The behavior of PAR, which seems to induce
very high spreads, is surprising. This behavior is because
we measure the market bid and ask only after the extraneous LMSR agent has intervened and perhaps cleared some
orders which would still be waiting in the books in the absence of a market maker, so the spread looks artficially large,
compared with pureCDA. In addition, PAR doesn’t actually
place any new orders on the books, since it waits for orders
to arrive before acting, as opposed to INT, which proactively
improves spreads by adding to the order book. This finding,
which casts doubt on the meaningfulness of spread measurement for PAR, is problematic since many real-life traders use
the spread to gauge market quality and make decisions.
To get a better idea of the market maker’s role in improving trading activity, we also computed the actual volume of
trade executed. We did this in two ways: for each simulation, we maintained a ledger where each entry recorded the
buyer, seller, execution price, and quantity of every market
trade; after ntrades episodes, we added all these traded quantities together to obtain Vol=quantity absorbed by buyers
and market maker (if present)=quantity supplied by sellers
and market maker. PAR beats both pureCDA and INT with
respect to this measure.
We also calculated an alternative measure of trading volume by subtracting the total residual quantity on the order
books at the end of each simulation from the total quantity ordered by all traders: Vol⇤ = quantity absorbed by
buyers (from sellers and market maker) + quantity supplied
by sellers (to buyers and market maker). It double-counts,
perhaps appropriately, every quantity traded between background traders, and thus reflects the overall “satisfaction”
of the entire background trader population in a way that
the previous measure does not. 3 Strangely, for higher variability in trader beliefs, PAR gives the worst Vol⇤ bettered
by INT and pureCDA, but there is a complete reversal in
this behavior as the variability decreases. Based on observations of some sample trade ledgers and order book residuals, we believe that the reason is this: in any CDA with
a market maker, the market maker gets the advantage of
immediacy due to its continuous presence and itself undercuts some of the background traders, thereby reducing the
(double-counted) quantity that changes hands between these
traders. Hence pureCDA, where every trade must occur between background traders, has a higher Vol⇤ . But with increasing Ntrials as trader beliefs get closer to each other, relatively more traders trade with other background traders, who
now offer competitive prices themselves. This is an interesting example of how the presence of the market maker can
affect the dynamics of trade in surprising ways.
Also note that regardless of the microstructure, both Vol
and Vol⇤ decrease as the knowledge of the trading crowd
gets more and more precise, which is consistent with the
idea that as the noise in the beliefs of traders with a common
knowledge structure reduces, trading becomes less profitable, hence less likely.
Welfare: Trader surplus or (weighted) price improvement
decreases with increasing precision in beliefs but the presence of a market maker consistently improves the surplus as
opposed to having only a CDA, PAR more so than INT. It
is also noteworthy that the combination of CDA and market making performs better in this respect than each of them
individually. Moreover, we consistently observe INT loss <
PAR loss ⇡ pureLMSR loss, and these losses respect the
known LMSR loss bound of 69.3 (starting price = 0.5,
B = 100). This empirical observation supports the notion
that Proposition 1 (which shows that myopic costs faced by
the market maker are lower for INT than for PAR when they
start from the same state in terms of q and the order books)
might generalize to expected losses over sequences of trades
from a particular starting point, an interesting direction for
theoretical work on the topic (in a handful of our individual simulations, INT made slightly more loss than PAR,
which shows that the sequence result cannot hold deterministically).
Discussion
We have introduced a new LMSR-based market making algorithm that applies to a CDA setting, and have compared its
properties with three other market microstructures in simulations with basic trading agents. Future research directions
include analyzing these market settings with more sophisticated trader models.
3
Of course, for pureLMSR, Vol⇤ =Vol since the market maker
takes one side of every trade.
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